Novel Models for Multiple Dependent Heteroskedastic Time Series

Functional magnetic resonance imaging or functional MRI (fMRI) is a very popular tool used for differing brain regions by measuring brain activity. It is affected by physiological noise, such as head and brain movement in the scanner from breathing, heart beats, or the subject fidgeting. The purpose of this paper is to propose a novel approach to handling fMRI data for infants with high volatility caused by sudden head movements. Another purpose is to evaluate the volatility modelling performance of multiple dependent fMRI time series data. The models examined in this paper are AR and GARCH and the modelling performance is evaluated by several statistical performance measures. The conclusions of this paper are that multiple dependent fMRI series data can be fitted with AR + GARCH model if the multiple fMRI data have many sudden head movements. The GARCH model can capture the shared volatility clustering caused by head movements across brain regions. However, the multiple fMRI data without many head movements have fitted AR + GARCH model with different performance. The conclusions are supported by statistical tests and measures. This paper highlights the difference between the proposed approach from traditional approaches when estimating model parameters and modelling conditional variances on multiple dependent time series. In the future, the proposed approach can be applied to other research fields, such as financial economics, and signal processing. Code is available at \url{https://github.com/13204942/STAT40710}.